An unbiased Ito type stochastic representation for transport PDEs: A Toy Example

TL;DR

This paper introduces an unbiased stochastic method for solving transport PDEs using Ito representations, leveraging recent advances in branching diffusions and regime switching, which outperforms existing approaches and extends to nonlinear cases.

Contribution

The paper presents a novel unbiased stochastic representation for transport PDEs that does not rely on small diffusion coefficients, enabling application to nonlinear PDEs where traditional methods fail.

Findings

Algorithm outperforms alternative approaches in examples

Method extends to nonlinear PDEs with first derivative terms

No reliance on smallness of diffusion coefficient

Abstract

We propose a stochastic representation for a simple class of transport PDEs based on Ito representations. We detail an algorithm using an estimator stemming for the representation that, unlike regularization by noise estimators, is unbiased. We rely on recent developments on branching diffusions, regime switching processes and their representations of PDEs. There is a loose relation between our technique and regularization by noise, but contrary to the latter, we add a perturbation and immediately its correction. The method is only possible through a judicious choice of the diffusion coefficient $\sigma$. A key feature is that our approach does not rely on the smallness of $\sigma$, in fact, our $\sigma$ is strictly bounded from below which is in stark contrast with standard perturbation techniques. This is critical for extending this method to non-toy PDEs which have nonlinear terms in the first derivative where the usual perturbation technique breaks down. The examples presented show the algorithm outperforming alternative approaches. Moreover, the examples point toward a potential algorithm for the fully nonlinear case where the method of characteristics breaks down.